SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels

Abstract

We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant of deep neural networks for irregular structured and geometric input, e.g., graphs or meshes. Our main contribution is a novel convolution operator based on B-splines, that makes the computation time independent from the kernel size due to the local support property of the B-spline basis functions. As a result, we obtain a generalization of the traditional CNN convolution operator by using continuous kernel functions parametrized by a fixed number of trainable weights. In contrast to related approaches that filter in the spectral domain, the proposed method aggregates features purely in the spatial domain. In addition, SplineCNN allows entire end-to-end training of deep architectures, using only the geometric structure as input, instead of handcrafted feature descriptors.For validation, we apply our method on tasks from the fields of image graph classification, shape correspondence and graph node classification, and show that it outperforms or pars state-of-the-art approaches while being significantly faster and having favorable properties like domainindependence. Our source code is available on GitHub 1 .Recently, a set of methods brought together under the term geometric deep learning [3] emerged, which aim to achieve this transfer by defining convolution operations for deep neural networks that can handle irregular input data. Existing work in this field can loosely be divided into two different subsets: the spectral and the spatial filtering approaches. The former is based on spectral graph theory [5], where eigenvalues of a graph's Laplacian matrix are interpreted as frequencies of node signals [22]. They are filtered in the spectral domain, analogously to Fourier domain filtering of traditional signals. The latter subset, the spatial approaches, perform convolution in local Euclidean neighborhoods w.r.t. local positional relations between points, represented for example as polar, spherical or Cartesian coordinates, as shown as examples in Figure 1.Contribution. We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant of deep neural networks for irregular structured data. The main contribution is a trainable, spatial, continuous convolution kernel that leverages properties of B-spline bases to efficiently filter geometric input of arbitrary dimensionality. We show

SpiralNet++: A Fast and Highly Efficient Mesh Convolution Operator

“…In contrast to previous approaches [14,28,34] which aggregate neighboring node features based on trainable weight functions, our method encodes node features under a explicitly defined spiral sequence, and a fully connected layer follows to encode input features combined with ordering information. It is a simple yet efficient approach.…”

Graph-Based Object Classification for Neuromorphic Vision Sensing

“…However, this operation requires identical graph input and handles the whole graph simultaneously, so it is not suitable for the variable and large graphs constructed from NVS. On the other hand, spatial convolution [20,38] aggregates a new feature vector for each vertex, us-ing its neighborhood information weighted by a trainable kernel function. Because of this property, we consider spatial convolution operation as a better choice when dealing with graphs from NVS.…”

“…The content of U is used to determine how the features are aggregated and the content of f l (j) defines what is aggregated. As such, several spatial convolution operations [20,38,40] on graphs were proposed by using different choice of kernel functions. Among them, SplineCNN [20] achieves state-of-the-art results in several applications, so in our work we use the same kernel function as in SplineCNN.…”

“…As such, several spatial convolution operations [20,38,40] on graphs were proposed by using different choice of kernel functions. Among them, SplineCNN [20] achieves state-of-the-art results in several applications, so in our work we use the same kernel function as in SplineCNN. In this way, we leverage properties of B-spline bases to efficiently filter NVS graph inputs of arbitrary dimensional- [47].…”

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“…as applied to harmonic analysis on graphs (Kotzagiannidis and Dragotti (2019)). More recently, Fey et al (2018) have proposed tensor B-splines defined over a Cartesian product basis for geometric convolutional neural networks. Kronecker sums have been proposed as precision matrices for weighting the quadratic regularizer in smoothed multivariate spline regression.…”